Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2

Average Error: 0.3 → 0.3

Time: 24.6s

Precision: binary64

Cost: 26304

Math

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

↓

\[\log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right) \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))

↓

(FPCore (x y z t a)
 :precision binary64
 (+ (log (+ x y)) (- (fma (+ a -0.5) (log t) (log z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	return ((log((x + y)) + log(z)) - t) + ((a - 0.5) * log(t));
}

↓

double code(double x, double y, double z, double t, double a) {
	return log((x + y)) + (fma((a + -0.5), log(t), log(z)) - t);
}

Julia

function code(x, y, z, t, a)
	return Float64(Float64(Float64(log(Float64(x + y)) + log(z)) - t) + Float64(Float64(a - 0.5) * log(t)))
end

↓

function code(x, y, z, t, a)
	return Float64(log(Float64(x + y)) + Float64(fma(Float64(a + -0.5), log(t), log(z)) - t))
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(N[(N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision] + N[(N[(a - 0.5), $MachinePrecision] * N[Log[t], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := N[(N[Log[N[(x + y), $MachinePrecision]], $MachinePrecision] + N[(N[(N[(a + -0.5), $MachinePrecision] * N[Log[t], $MachinePrecision] + N[Log[z], $MachinePrecision]), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]

Target

Original	0.3
Target	0.3
Herbie	0.3

\[\log \left(x + y\right) + \left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right) \]

Derivation

1. Initial program 0.3

   \[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]

2. Simplified 0.3

   \[\leadsto \color{blue}{\log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right)} \]
   Proof

   [Start] 0.3	\[ \left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \left(a - 0.5\right) \cdot \log t \]
   associate-+l- [=>] 0.3	\[ \color{blue}{\left(\log \left(x + y\right) + \log z\right) - \left(t - \left(a - 0.5\right) \cdot \log t\right)} \]
   associate--l+ [=>] 0.3	\[ \color{blue}{\log \left(x + y\right) + \left(\log z - \left(t - \left(a - 0.5\right) \cdot \log t\right)\right)} \]
   associate-+l- [<=] 0.3	\[ \log \left(x + y\right) + \color{blue}{\left(\left(\log z - t\right) + \left(a - 0.5\right) \cdot \log t\right)} \]
   +-commutative [=>] 0.3	\[ \log \left(x + y\right) + \color{blue}{\left(\left(a - 0.5\right) \cdot \log t + \left(\log z - t\right)\right)} \]
   associate-+r- [=>] 0.3	\[ \log \left(x + y\right) + \color{blue}{\left(\left(\left(a - 0.5\right) \cdot \log t + \log z\right) - t\right)} \]
   fma-def [=>] 0.3	\[ \log \left(x + y\right) + \left(\color{blue}{\mathsf{fma}\left(a - 0.5, \log t, \log z\right)} - t\right) \]
   sub-neg [=>] 0.3	\[ \log \left(x + y\right) + \left(\mathsf{fma}\left(\color{blue}{a + \left(-0.5\right)}, \log t, \log z\right) - t\right) \]
   metadata-eval [=>] 0.3	\[ \log \left(x + y\right) + \left(\mathsf{fma}\left(a + \color{blue}{-0.5}, \log t, \log z\right) - t\right) \]

3. Final simplification 0.3

   \[\leadsto \log \left(x + y\right) + \left(\mathsf{fma}\left(a + -0.5, \log t, \log z\right) - t\right) \]

Alternatives

Alternative 1
Error	12.6
Cost	20041

\[\begin{array}{l} \mathbf{if}\;a \leq -1.45 \lor \neg \left(a \leq 1.95\right):\\ \;\;\;\;\log \left(x + y\right) + \left(a \cdot \log t - t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\log z + \left(\log y + -0.5 \cdot \log t\right)\right) - t\\ \end{array} \]

Alternative 2
Error	0.3
Cost	20032

\[\left(\left(\log \left(x + y\right) + \log z\right) - t\right) + \log t \cdot \left(a + -0.5\right) \]

Alternative 3
Error	12.1
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\left(\log z + \log y\right) + \log t \cdot \left(a + -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 4
Error	12.1
Cost	19908

\[\begin{array}{l} \mathbf{if}\;t \leq 3.75 \cdot 10^{-6}:\\ \;\;\;\;\log z + \left(\log y + \log t \cdot \left(a + -0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot \log t - t\\ \end{array} \]

Alternative 5
Error	20.0
Cost	19904

\[\left(\log z - t\right) + \left(\log y + \log t \cdot \left(a + -0.5\right)\right) \]

Alternative 6
Error	11.3
Cost	14228

\[\begin{array}{l} t_1 := \log \left(\left(x + y\right) \cdot \left(z \cdot {t}^{\left(a + -0.5\right)}\right)\right) - t\\ t_2 := a \cdot \log t\\ t_3 := \left(\log z - t\right) + t_2\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{-63}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-209}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-247}:\\ \;\;\;\;\log \left(\left(z \cdot \left(x + y\right)\right) \cdot {t}^{-0.5}\right) - t\\ \mathbf{elif}\;a \leq 1.15:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_2 - t\right)\\ \end{array} \]

Alternative 7
Error	9.4
Cost	14032

\[\begin{array}{l} t_1 := \left(-0.5 \cdot \log t + \log \left(z \cdot \left(x + y\right)\right)\right) - t\\ t_2 := a \cdot \log t\\ t_3 := \log \left(x + y\right) + \left(t_2 - t\right)\\ \mathbf{if}\;a \leq -8.2 \cdot 10^{-12}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-209}:\\ \;\;\;\;\left(\log z - t\right) + t_2\\ \mathbf{elif}\;a \leq 2.25 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 8
Error	17.3
Cost	13904

\[\begin{array}{l} t_1 := \left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ t_2 := a \cdot \log t\\ t_3 := \log \left(x + y\right) + \left(t_2 - t\right)\\ \mathbf{if}\;a \leq -1.15 \cdot 10^{-11}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-185}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\left(\log z - t\right) + t_2\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 9
Error	15.6
Cost	13904

\[\begin{array}{l} t_1 := a \cdot \log t\\ t_2 := \left(\log z - t\right) + t_1\\ \mathbf{if}\;a \leq -3.6 \cdot 10^{-62}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -4.6 \cdot 10^{-185}:\\ \;\;\;\;\log \left(\left(z \cdot \left(x + y\right)\right) \cdot {t}^{-0.5}\right) - t\\ \mathbf{elif}\;a \leq -1.18 \cdot 10^{-209}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-11}:\\ \;\;\;\;\left(-0.5 \cdot \log t + \log \left(y \cdot z\right)\right) - t\\ \mathbf{else}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_1 - t\right)\\ \end{array} \]

Alternative 10
Error	16.3
Cost	13772

\[\begin{array}{l} t_1 := \log \left(y \cdot z\right) + \log t \cdot \left(a + -0.5\right)\\ t_2 := a \cdot \log t\\ \mathbf{if}\;t \leq 7.5 \cdot 10^{-205}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{-131}:\\ \;\;\;\;\log \left(x + y\right) + \left(t_2 - t\right)\\ \mathbf{elif}\;t \leq 2.4 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\log z - t\right) + t_2\\ \end{array} \]

Alternative 11
Error	14.6
Cost	13376

\[\log \left(x + y\right) + \left(a \cdot \log t - t\right) \]

Alternative 12
Error	14.9
Cost	13248

\[\left(\log z - t\right) + a \cdot \log t \]

Alternative 13
Error	24.3
Cost	6724

\[\begin{array}{l} \mathbf{if}\;t \leq 3.15 \cdot 10^{+35}:\\ \;\;\;\;a \cdot \log t\\ \mathbf{else}:\\ \;\;\;\;-t\\ \end{array} \]

Alternative 14
Error	16.5
Cost	6720

\[a \cdot \log t - t \]

Alternative 15
Error	39.7
Cost	128

\[-t \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z t a)
  :name "Numeric.SpecFunctions:logGammaL from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (+ (log (+ x y)) (+ (- (log z) t) (* (- a 0.5) (log t))))

  (+ (- (+ (log (+ x y)) (log z)) t) (* (- a 0.5) (log t))))